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Niels Koch
von Koch

Koch Curve

"It is this similarity between the whole and its parts, even infintesimal ones, that makes us consider this curve of von Koch as a line truly marvelous among all. If it were gifted with life, it would not be possible to destroy it without anihilating it whole, for it would be continually reborn from the depths of its triangles, just as life in the universe is."

E. Cesaro, Atti d. R. Accademia d. Scienze d. Napoli, 2, XII, number 15. [Excerpt from an article by Paul Lévy, reprinted in Edgar's text Classics on Fractals.]

Begin with a straight line (the blue segment in the top figure). Divide it into three equal segments and replace the middle segment by the two sides of an equilateral triangle of the same length as the segment being removed (the two red segments in the middle figure). Now repeat, taking each of the four resulting segments, dividing them into three equal parts and replacing each of the middle segments by two sides of an equilateral triangle (the red segments in the bottom figure). Continue this construction.

The Koch curve is the limiting curve obtained by applying this construction an infinite number of times. For a proof that this construction does produce a "limit" that is an actual curve, i.e. the continuous image of the unit interval, see the text by Edgar.

The first iteration for the Koch curve consists of taking four copies of the original line segment, each scaled by r = 1/3. Two segments must be rotated by 60°, one counterclockwise and one clockwise. Along with the required translations, this yields the following IFS

The fixed invariant set of this IFS is the Koch curve.

Angle 60
Axiom F
F --> F+F- -F+F
We have a hyperbolic IFS with each map being a similitude of ratio r < 1. Therefore the similarity dimension, d, of the unique invariant set of the IFS is the solution to

Koch constructed his curve in 1904 as an example of a non-differentiable curve, that is, a continuous curve that does not have a tangent at any of its points. Karl Weierstrass had first demonstrated the existence of such a curve in 1872. The article by Sime Ungar provides a simple geometric proof.

The length of the intermediate curve at the nth iteration of the construction is (4/3)^n, where n = 0 denotes the original straight line segment. Therefore the length of the Koch curve is infinite. Moreover, the length of the curve between any two points on the curve is also infinite since there is a copy of the Koch curve between any two points.

Three copies of the Koch curve placed around the three sides of an equilateral triangle, form a simple closed curve that form the boundary of the Koch snowflake.

 
  1. Bannon, Thomas. "Fractals and Transformations," Mathematics Teacher, March 1991.
  2. Barcellos, Anthony. "The Fractal Geometry of Mandelbrot," College Mathematics Journal.
  3. Camp, Dane. "A Fractal Excursion," Mathematics Teacher, April 1991, 265-275.
  4. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Sringer-Verlag, 1990.
  5. Edgar, Gerald A. Classics on Fractals, Addison-Wesley 1993. Contains a translation of Koch's original article (see [8]).
  6. Gardner, Martin. "Mathematical Games," Scientific American, April 1965, 128.
  7. Jones, Juw. "Fractals Before Mandelbrot-A Selective History," in Fractals and Chaos, Crilly, Earnshaw, and Jones, Editors, Springer-Verlag 1991.
  8. Koch, H. von. "Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire," Arkiv for Matematik 1 (1904) 6810704.
  9. Mandelbrot, Benoit. The Fractal Geometry of Nature, W.H. Freeman and Co. 1983.
  10. McWorter Jr., William A. and Jane Morrill Tazelaar. "Creating Fractals,", Byte, August 1987, 123-132.
  11. Peitgen, Heinz-Otto, Hartmut Jurgens and Dietmar Saupe. Fractals for the Classroom, Part One: Introduction to Fractals and Chaos, Springer-Verlag New York, Inc. 1990.
  12. Ungar, Sime. "The Koch Cuve: A Geometric Proof," The American Mathematical Monthly, Vol. 114, No. 1 (January 2007), 61-66.
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